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In his comparison of poetry to history, Aristotle points out that their difference is not one between verse and prose. After all, the writings of Herodotus would be a species of history with meter no less than without it. The real difference that distinguishes them is between telling what might be and what has been. Notoriously, the Stagirite takes this distinction for the reason why poetry is more scientific [philosophôteron] and more serious [spoudaioteron] than history. For poetry tells of general truths, which is the sort of thing that a certain type of man will do or say either probably or necessarily [to eikos ê to anankaion]. By contrast, history tells of particular facts such as what Alcibiades did or suffered [epraxen ê ti epathen].[i]

Yet as Aristotle inaugurates philosophy with his account of general truths pertaining to the words and deeds of a certain type of man, his teacher Plato by these lights counts for no less of a poet without meter, than he might have appeared with it. Let us bear in mind Aristotle’s contrast in the following exploration of two poetic archetypes, Socrates and Gorgias, in their dramatic debate about the virtues of rhetoric. Their wrangle recorded in Plato’s Gorgias, and the techniques of Thucydides’ History of the Peloponnesian War, will ground this inquiry into the relation of rhetoric to reason.

2.

Myles Burnyeat summarized his account of Aristotle on the rationality of rhetoric in the form of question and answer: “We would like to know under what conditions it is appropriate for a speaker to advance, and for the audience to accept, a sign argument that is deductively invalid? The only answer we get from the Rhetoric is: when it is convincing.”[ii] For Socrates, this answer will not do. Nothing short of certainty will satisfy him. He engages in arguments by alternating between the roles of the speaker and his audience. He aims to reveal hitherto unrecognized errors to his interlocutors, by guiding them to infer contradictions from their theses or to deduce their antitheses. He has no use for conviction unwarranted by indisputable demonstration. In Socratic dialectic, only valid arguments are worth being advanced and accepted, and their advancement and acceptance are warranted only in the pursuit of just ends. Socrates conveys his dialectic reasoning through a technique of maieutics, his service to his interlocutors’ ideas being a counterpart to a midwife assisting childbirth. He coaxes conscious understanding from latent ideas in the course of a dialogue conducted as a series of pointed questions and brief answers. The progress of this dialogue depends on achieving unshakeable consensus on each successive point. This elenctic protocol allows discovery through reconciling or choosing between competing viewpoints. By following it, Socrates aims to achieve mutual understanding through stepwise accrual of agreement. In the ideal case, a mathematical proof ensures absolute certainty.[iii]

In his historical conduct, Gorgias the sophist neither restricted his means of persuasion to demonstrative, geometric reasoning, nor imposed moral constraints on its aims.[iv] Although in his Apology, Plato has Socrates name Gorgias of Leontini alongside Prodicus of Ceos and Hippias of Eos as sophists, or commercial purveyors of wisdom, the eponymous character in Gorgias modestly identifies himself as a rhetorician, in setting out to praise the role of rhetoric in society.[v] Associating his trade with liberty and power in a democracy, he defines rhetoric as an art of speeches [logoi] that aim to produce persuasion regarding the just and unjust.[vi] Rhetorical speeches are about the greatest and the best human affairs, which is the cause of freedom for men and the basis of rule over others in their city. They are equally fit to persuade judges in a law court, senators in the Council chamber, assemblymen in the Assembly, and the multitude in common political gatherings.[vii]

In fact, rhetorical ability counted for a great deal in the functioning of Athenian democracy. Most men active in politics sought training, and vied for recognition, as orators. In the best public venues, rhetoric was recognized as the discipline most suited for directing human affairs.[viii] But high demand inspired suspicion. By stressing the nature of rhetoric as an instrument of persuasion, Gorgias lays himself open to the charge that rhetoric aims at belief without knowledge. His examples of Themistocles and Pericles aggravate this weakness. The Long Walls were built to link Athens securely to its harbors at Piraeus and Phalerum. The passage they secured ensured that the city could not be encircled by an invading army and besieged by land alone. After the Persian Wars reduced them to rubble, Sparta pressed Athens to stop rebuilding her walls, lest they create a base for another Persian invasion. But advocacy by Themistocles and Pericles eventually caused their reconstruction.[ix] These politicians employed their rhetorical powers to advise Athenians on building their walls; yet they were neither architects nor stonemasons.[x] Thus Socrates turns Gorgias’ example against its maker, who had disclaimed orators’ need to know how things really stand with things themselves, requiring them only to discover some trick of persuasion, so as to appear to the unknowing to know more than those who know.[xi] Rebutting this claim, Socrates suggests that in employing their rhetorical powers, these politicians aimed only at accommodating people’s appetites [epithumiai]. He neglects to point out that in promoting public works, Themistocles had to argue against distributing their budget among the people. Nonetheless, he succeeds in establishing that in the long term Pericles was impelled by agenda to ratify and satisfy the desires of his constituents rather than guide them towards moral improvement.[xii]

In regard of this moral concern, Gorgias volunteers a critical concession, that rhetoric should not be used indiscriminately against any target, any more so than the fighting arts should be used against friend and foe alike.[xiii] Nevertheless, he goes on to claim that his universal art allows him to surpass experts in their disciplines, that he can persuade the average man to take a stand in any area of knowledge, and that he can do all that without having to learn anything of particular substance.[xiv] Arguing against this thesis, Socrates compels the rhetorician to concede that he both knows the nature of the good and bad, the fine and the shameful, the just and unjust, and places himself in the right regarding each moral distinciton.[xv] At this point Gorgias has committed himself to a fatal contradiction. His admissions imply that the rhetorician must know and respect all moral qualities, while falling short of the capacity to teach them to his students. He shares the philosopher’s knowledge, but not his ability to communicate it. The historical Gorgias was credited with proving three remarkable propositions: that nothing exists; that even if it does exist, it is incomprehensible to man; and that, even if it is comprehensible to anyone, it is not communicable to anyone else.[xvi] A sophist of this caliber would not have been embarrassed by having to profess non-communicable knowledge. But the dignified rhetorician respectfully portrayed within Plato’s dialogue concedes the game for want of sophistical shamelessness, entitling Socrates to conclude that the rhetorician is a manufacturer of groundless belief, and condemn oratory as no art [technê], but a mere knack, a species of flattery altogether lacking in dignity.[xvii]

3.

Unlike the characters of Plato’s dialogue, Aristotle identifies the technical nature of his subject matter in the Rhetoric as the counterpart of dialectic. The Aristotelian speaker advances his argument through a process of proof. He presents considerations regarding his subject, drawing upon all available premisses to reach the desired conclusion, whilst anticipating the objections of his audience. He strives to compel his audience into accepting a convincing argument to bear on its future decisions. Although rhetoric and dialectic both deal with matters that concern all human understanding, they differ in their means of demonstration. A rhetorical argument proceeds from received opinions[endoxa], leaving plenty of wiggle room for filling the gaps in their demonstration. In contrast to rhetoricians, dialecticians’ reasoning proceeds from premisses accepted by their respondents via arguments that their respondents recognize as logically valid. Socratic arguments require reasoned discussion with no room for objection. But whereas the dialectic technique of maieutics only allows a proceeding after a consensus is made, each rhetorical debate remains open to challenge at every step, ruling out conclusive arguments in perpetuity.

Aristotle blames his predecessors for saying nothing about enthymemes that belong to the body of proof, but chiefly devoting their attention to matters outside the subject; for the arousing of prejudice, compassion, anger, and similar emotions having no connexion with the matter in hand, but directed only to the dicast charged with deciding their case. Thus in his account of rhetoric Aristotle avoids both the Gorgian praise and the Socratic condemnation. Though his technique aims to convince through the motion of affects, proofs comprise its only aspect that comes within the province of art, everything else being merely an accessory. Enthymemes are the body of proof.[xviii] Accordingly, in order to understand the nature of proof, we must pin down the nature of enthymemes.

An enthymeme is a sort of argument [sullogismos tis] used in a rhetorical speech.[xix] Its material is derived from four sources, likelihood [eikos], example [paradeigma], necessary sign [tekmêrion], and sign [sêmeion]. Only enthymemes based on necessary signs [tekmêria], can lead to conclusions that are beyond refutation.[xx] But in the general case, these is no need to preempt the possibility of refutation. The rhetorician aims instead to establish his case to the best of his ability, proving it to the satisfaction of an audience [pistis].[xxi] Thus, besides enthymemes, amplifications and examples are admissible techniques for proof:

Speaking generally, of the topics common to all rhetorical arguments, amplification is most suitable for epideictic speakers, whose subject is actions which are not disputed, so that all that remains to be done is to attribute beauty and importance to them. Examples are most suitable for deliberative speakers, for it is by examination of the past that we divine and judge the future. Enthymemes are most suitable for forensic speakers, because the past, by reason of its obscurity, above all lends itself to investigation of causes and to demonstrative proof.[xxii]

It is clear that the aim of Aristotelian rhetoric far exceeds the exiguous means of geometrical demonstration. Thus hyperbole has a place in declamations that take bare facts as undisputed. Likewise, examples that support the contested proposition inductively can be taken as the basis for sustaining it as a probable generalization from particular instances. In practice, such proof succeeds whenever it can sway the audience into making its decisions on the most probable ground. But probability will vary depending on the circumstances. And in cases that fall short of certainty, the rhetorician can only hope and pray that his audience includes no rational detectors of error capable of deriving a contradiction from his thesis or formulating the proof of its antithesis.

By Aristotle’s lights, Socrates’ reasoning in his debate with Gorgias may be faulted for a gratuitous dichotomy, an unwarranted division of a whole into two mutually exclusive parts. Socrates presents to Gorgias with two mutually exclusive choices, implicitly ruling out any unstated alternatives. On the one side stand philosophers and physicians, teachers and artisans. On the other side congregate flatterers and suckers, demagogical politicians and ignorant multitudes. As Socrates claims his place among the former honest and forthright folk, he classifies Gorgias among the latter ilk, purveyors and consumers of baseless belief and unsound fodder. However, must every politician only aim towards gratifying his constituents? Surely Themistocles and Pericles did not have to instruct Athenians in the art of masonry in order to convince them of the importance of building the wall. Their proposals legitimately relied on division of labor that ensured full participation of builders in public debates. Freed thereby from technical concerns, the politicians were right to focus on ensuring security for their constituents. Likewise, as an expert in persuasion, Gorgias should have been able to team up with experts in any discipline related to its subject matter in any particular instance. But even in his modest purview of Plato’s dialogue, the rhetorician is not modest enough to disclaim self-sufficiency. This failure needlessly foredooms his confrontation with the philosopher.

4.

Within the historical perspective, Aristotelian criticism on Socrates and Gorgias finds a basis in Thucydides’ History of the Peloponnesian War. Paul Shorey aptly characterized Thucydides as “a hard-headed […] rationalist who was contemptuous of all teleological and providential interpretations of history and explained everything by natural causes and unchanging human nature—the psychology, motives, and the conflicting interests of men.”[xxiii] Tradition contrasts this portrayal of Thucydides with the received image of Herodotus via a backhanded compliment. Herodotus, simultaneously anointed as the father of history and disparaged as the father of lies, lays himself open to criticism as a casual entertainer, if not outright denunciation as an irresponsible fantasist.[xxiv] Whereas the paternity of scientific history allotted to Thycydides in recognition of his analytical rigor, contains in its technical qualification the gloomy image of a mechanistic skeleton propelled by spasms of cynicism through a morass of tedium. Nonetheless, it would be wrong to withhold credit for Thucydides anticipating the Aristotelian treatment of proof, albeit in a way that conflated probable proof [sêmeion] with necessary proof [tekmêrion].[xxv] This conflation addressed his concern and indicated the way he sought to resolve:

For though the events of remote antiquity, and even those that more immediately precede the war, could not from lapse of time be clearly ascertained, yet the evidences which an inquiry carried as far back as was practicable leads me to trust, all point to the conclusion that there was nothing on a great scale, either in war or in other matters.[xxvi]

Even when the historical facts are obscured by the passage of time, available evidence is the key to inferring their contours. However, in composing his account, Thucydides structures all particular evidence in accordance with the dictates of general principle. Thus the rhetorical arguments in the speeches that Thucydides incorporated in his account of the Peloponnesian war anticipate Aristotle’s idea of rhetorical proof, in being based on the most reputable signs and connecting with the concerns of its audience. Although the composition of each speech is grounded in specific evidence from each individual event, its first allegiance is to the intrinsic logic of their makers’ circumstances:

With reference to the speeches in this history, some were delivered before the war began, others while it was going on; some I heard myself, others I got from various quarters; it was in all cases difficult to carry them word for word in one’s memory, so my habit has been to make the speakers say what was in my opinion demanded of them by the various occasions, of course adhering as closely as possible to the general sense of what they really said.[xxvii]

Thucydides amassed and dispensed historical knowledge not for its own sake, but as a conduit to understanding. Far from resting content in accounts of particular facts such as what men did or what was done to them, he aimed to uncover and convey general truths about human action. Beyond establishing the patterns of masses in turmoil and plots of demagogues clinging to power, his history aimed at dissecting the nature of social upheavals and unmasking demagoguery, indeed at penetrating political power itself. In accounting for moral and political issues, his main device was the speech. The purpose of the ensuing historical writing is to guide its readers toward an understanding of actions and events as determined by the energies that impel human agents and forces that constrain them. Its allegiance to conclusions borne out by factual evidence checked this speculative urge. Thus Thucydides would punctuate factually grounded interpretation, rendered more plausible by his impersonal tone, with spells of invention that attributed discourses to his characters.[xxviii]

The Mytilene debate in Book III is an example of proof presented through the twin means of impersonal narrative and revealing speeches, which are equally embedded into their context.[xxix] The debate takes place on the day following the order for total extermination of the Mytilene men and enslavement of their women and children, agreed upon by the Athenian assembly and dispatched to Mytilene. Thucydides introduces it by noting repentance and reflection on the cruelty of a decree that condemned a whole city to the fate merited only by the guilty, which caused a second assembly to be summoned.[xxx] Both of the following speeches present their makers’ arguments with proofs that illustrate possible consequences and anticipate the audience’ thoughts so as to guide it towards a decision. Cleon argues for executing the original order. He intends this extreme course of action to seal the Athenian victory and forestall future conflicts by deterring other cities from revolt. Mytilene should not have had a chance to build up their arrogance for attack. The right response to their revolt must deter all remaining allies from breaking faith with Athens. Athenians should not let themselves be swayed by clever speeches or large bribes. The penalty for rebellion is death[xxxi] In his response, Diodotus argues from the opposite position, advocating execution only for the leaders of the rebels. He disclaims any motive in regard to the Mytilenians, besides the reasons of state: “Though I prove them ever so guilty, I shall not, therefore, advise their death, unless it be expedient; nor though they should have claims to indulgence, shall I recommend it, unless it be clearly for the good of the country.”[xxxii] He stresses that the discussion should concern the present rather than the future. Athenians should think how Mytilene could be most useful to their polis. Their death would not deter others from breaking laws. On the contrary, it would inspire any future rebels to rule out surrender and fight to the death. A harsh penalty would increase future losses. In dealing with free people, Athens should favor timely prevention over belated punishment, taking tremendous care of them to forestall the mere idea of their revolt.[xxxiii]

Both discourses urge their audience to resist emotions that might sway their rational judgment. Cleon speaks of Mytilenians who had forfeited their right to be pitied by the Athenians in virtue of having rebelled against them. Men should extend their sympathy to friends, not to enemies. He warns the audience against falling prey to their own pleasure in considering the opposite view. Diodotus opens his response by identifying “the two things most opposed to good counsel [as] haste and passion; haste usually goes hand in hand with folly, passion with coarseness and narrowness of mind.”[xxxiv] Thus he responds to the bias towards anger at the Mytilenes’ revolt that would incline his audience to agree with the policy of total extermination and enslavement. Diodotus directs his audience toward their interests in the situation. In this regard, Thucydides’ construction of proof anticipates the rhetoric of Aristotle. His speakers appear to forgo emotional appeals to their audience, concentrating instead on their interests. In the terms of Aristotle’s contrast in the Poetics, they argue as poets, not as historians. But surely this title belongs to the author, in his capacity of the puppetmaster of his characters.

5.

Thus historical arguments depend on uncertainty of actions and events, involving probability as a necessary quality in proof and leaving room for doubt in all future discussions. But there remains a Socratic tradition that seeks geometrical certainty in all matters. Between 1274 and 1305, Ramón Llull envisioned his Ars Magna as a system of mechanical means capable of drawing upon the totality of concepts so as to exhaust all combinatorial alternatives of their logical aggregation. Three and a half centuries later, in the first part of his 1655 treatise De Corpore, entitled “Computatio sive Logica” and intended as an introduction to his entire philosophical system, Thomas Hobbes speculated that the first truths “were arbitrarily made by those that first of all imposed Names upon Things, or received them from the imposition of others.” By this conventionalist approach to the necessary truths of mathematics, Hobbes distinguished Euclid’s axioms from the laws of physics, which are not made by arbitrary definitions. But even as he placed himself outside of its Platonist purview, Hobbes continued the project of Llull by treating human thought as reducible to the manipulation of signs, as a species of calculation.

In 1666, inspired by the analysis of Hobbes, 19-year old Gottfried Wilhelm Leibniz wrote his Dissertatio de Arte Combinatoria, envisioning the characteristica universalis, a method for precise resolution of all human disagreements. He speculated elsewhere that if we had it, we should be able to reason in metaphysics and morals in much the same way as in Geometry and Analysis, because the Symbols would clarify our thoughts that are too vague and too flighty in these matters, where imagination does not help us, if it would not do so through symbols:

Quo facto, quando orientur controversiae, non magis disputatione opus erit inter duos philosophos, quam inter duos Computistas. Sufficiet enim calamos in manus sumere sedereque ad abacos, et sibi mutuo (accito si placet amico) dicere: calculemus.If controversies were to arise, there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pencils in their hands, and say to each other (with a friend as witness, if they liked): Let us calculate.

Leibniz had no illusions about philosophical reasoning attaining the cogency of mathematical demonstration. There are no Euclidists and Archimedians in mathematics, as there are Aristotelians and Platonists in philosophy. Philosophers lack recourse to mathematical means of discovering possible mistakes. To that end, they require symbols and rules to formalize their thought and make it fit subject for calculation. The outcome of this procedure would endure in perpetuity, just as a mathematical truth, once understood, is never rejected.

Nonetheless, Leibniz acknowledged the limitations of his characteristica universalis. Its means could never suffice for deducing an individual statement like “Caesar was murdered on the ides of March”, because any such statement involves an infinity of causes and each of its constituent individual notions like Caesar comprises an infinity of elements. Nearly twenty years after inaugurating his program, Leibniz became even more skeptical about its prospects, observing that there are people who even reject indisputable arguments.[xxxv]

Leibniz’s empiricist foil John Locke approached the relationship between geometric demonstration and forensic persuasion from the opposite perspective:

As Demonstration is the shewing the Agreement, or Disagreement of two Ideas, by the intervention of one or more Proofs, which have a constant, immutable, and visible connexion one with another: so Probability is nothing but the appearance of such an Agreement, or Disagreement, by the intervention of Proofs, whose connexion is not constant and immutable, or at least is not perceived to be so, but is, or appears for the most part to be so, and is enough to induce the Mind to judge the Proposition to be true, or false, rather than the contrary. For example: In the demonstration of it, a Man perceives the certain, immutable connexion there is of Equality, between the three Angles of a Triangle, and those intermediate ones, which are made use of to shew their Equality to two right ones: and so, by an intuitive Knowledge of the Agreement, or Disagreement of the intermediate Ideas in each step of the progress, the whole Series is continued with an evidence, which clearly shews the Agreement, or Disagreement, of those three Angles, in equality to two right ones: And thus he has certain Knowledge that it is so. But another Man, who never took the pains to observe the Demonstration, hearing a Mathematician, a Man of credit, affirm the three Angles of a Triangle to be equal to two right ones, assents to it; i.e. receives it for true. In which case, the foundation of his Assent is the Probability of the thing, the Proof being such, as for the most part carries Truth with it: The Man, on whose Testimony he receives it, not being wont to affirm any thing contrary to, or besides his Knowledge, especially in matters of this kind. So that that which causes his Assent to this Proposition, that the three Angles of a triangle are equal to two right ones, that which makes him take these Ideas to agree, without knowing them to do so, is the wonted Veracity of the Speaker in other cases, or his supposed Veracity in this.[xxxvi]

Locke’s distinction suggests that appeal to probability differs from demonstrative reasoning in the fit to its audience. The speaker’s discretion is not only in following the injunction laid down near the beginning of Nicomachean Ethics, to achieve that amount of precision, which belongs to its subject matter,[xxxvii] but also in establishing the degree of certainty in proof that his audience requires and appreciates. Some of the most vital political matters that confront the American electorate today admit neither the utmost amount of precision nor the greatest degree of certainty. Mark Bowden articulates a case in point by recommending that torture should be banned but also quietly practiced:

In other words, when the ban is lifted, there is no restraining lazy, incompetent, or sadistic interrogators. As long as it remains illegal to torture, the interrogator who employs coercion must accept the risk. He must be prepared to stand up in court, if necessary, and defend his actions. Interrogators will still use coercion because in some cases they will deem it worth the consequences. This does not mean they will necessarily be punished. In any nation the decision to prosecute a crime is an executive one. A prosecutor, a grand jury, or a judge must decide to press charges, and the chances that an interrogator in a genuine ticking-bomb case would be prosecuted, much less convicted, is very small.[xxxviii]

The availability of the affirmative defense of necessity under common law defines the boundaries of precision and certainty in Anglo-American administration of criminal justice.[xxxix] It suggests that in the extreme circumstances, the best proof we can hope for in forensic arguments is the finding of reasonable doubt by a jury of our peers. Likewise our history has to content itself with provisional verdicts beyond reasonable doubt. As long as this state of affairs endures, the rationalist historian cannot hope to limit his demonstrations to valid arguments proceeding from true premisses.

6.

[…]

7.

As Georg Wilhelm Friedrich Hegel admitted in concluding the preface to his Philosophy of Right, philosophy always comes on the scene too late to give instruction as to what the world ought to be: “the owl of Minerva spreads its wings only with the falling of the dusk.” But the historian that follows Thucydides in poiesis, takes his cue from a different bird.

[i] Aristotle, Poetics1451a36-b11. I cite Aristotle by Bekker’s and Plato by Stephanus’ pagination. Whenever possible, I follow the Loeb editions and translations of classical texts, as available online at the Perseus Project. I thank Chien-Ling Liu for indispensable assistance with historical research and analysis.

[v] See Apology19e; compare the more attenuated characterization of Gorgias submitting himself to questioning by all comers on all subjects, including virtue, while disclaiming an ability to teach it, reported in Meno70b, 71c-d, 73c, 76b-c, 79e, 95c, and 96d.

[xii] See History of the Peloponnesian WarI, 90; compare the claims in History of the Peloponnesian WarII, 65. I am indebted for this point to the commentary in Plato, Gorgias, translated with notes by Terence Irwin, Oxford: Oxford University Press, 1979, p. 237.

[xix] See Rhetoric1355a4-7, 1400b37. I am equally indebted to the previously cited account of M.F. Burnyeat and its incisive criticism by Carlo Ginzburg in “Aristotle and History, Once More”, in History, Rhetoric, and Proof, Brandeis University Press, 1999, pp. 38-53. My understanding of enthymeme agrees with the traditional definition of an abbreviated syllogism, repudiated by Burnyeat and reinstated by Ginzburg.

[xxiv] See Cicero, De LegibusI.5, where Herodotus, acknowledged as the father of history, “pater historiae” is said to purvey find fables scarcely less numerous than those, which appear in the works of the poets; cf. the English translation by Francis Barham. Also see the discussion of the Herodotean and the Thucydidean traditions by Arnaldo Momigliano in The Classical Foundations of Modern Historiography, Berkeley: University of California Press, 1990, pp. 29-53, especially pp. 36-39 and 42-44.

[xxv] See e.g. his inference from persisting local customs to hypothetical past usage spread everywhere, in History of the Peloponnesian WarI.6, and other examples cited by Carlo Ginzburg in op. cit., pp. 44-45.

[xxxix] See A.W.B. Simpson, Cannibalism and the Common Law: The Story of the Tragic Last Voyage of the Mignonette and the Strange Legal Proceedings to Which It Gave Rise, Chicago: University of Chicago Press, 1984. The text of the judgment in the criminal case Regina v. Dudley and Stephens ([1884] 14 QBD 273 DC), establishing the precedent for the defense of necessity against criminal charges. Also see the hypothetical case described by Lon L. Fuller in “The Case of the Speluncean Explorers”, Harvard Law Review, Vol. 62, No. 4, February 1949.

Leibniz was well aware of the fact that our philosophical reasoning often lacks the cogency of mathematical reasoning (cf. [15, IV, p. 468]). There are no Euclidists and Archimedians in mathematics, as there are Aristotelians and Platonists in philosophy [16, ser. VI, vol. 4A, p. 695]. Whereas mathematicians have their own means of discovering possible mistakes, philosophers, who do not have such means at their disposal, should adopt rigorous reasoning all the more [16, ser. VI, vol. 4A, p. 705; ser. II, vol. 1, pp. 475, 478; 15, IV, p. 469]. Rationality should be such as to allow for a mathematisation of our thought; just as mathematicians have introduced letters and other symbols to designate mathematical objects and rules for operating with them, Leibniz proposed finding symbols and rules, in order to formalize a considerable part of our thought [20,10]. Then two philosophers with different opinions on a philosophical topic would no longer need to quarrel; they could say to each other “calculemus” (let’s calculate) [16, ser. VI, vol. 4A, p. 493]. Therefore, Leibniz’s invention of a calculating machine had a strong philosophical relevance. And besides, Leibniz tells us, this characteristica universalis would be an efficient means of converting pagans, because the true religion is the most rational religion, and it is impossible to resist rational arguments. In addition, apostasy will no longer occur, just as a mathematical truth, once understood, is never rejected. [Note 2: [16, ser. VI, vol. 4A, p. 269; ser. II, vol. I, p. 491]. Nearly twenty years later Leibniz was more skeptical: there are people who even reject indisputable arguments [16, ser. I, vol. 13, pp. 553-554].]

Leibniz made several efforts to find suitable symbols for the representation of our thinking. A very simple and nonetheless very interesting one was his idea that there are primary or irreducible notions and composite notions. If this is true, a map from notions to natural numbers can be defined, mapping primary notions to prime numbers and the relation “implies” between notions to the relation “is divisible by” between numbers. For illustration’s sake, Leibniz gives the example of the traditional definition of the human being as the rational living being. If the notion “rational” is mapped to the number 2 and the notion “being alive” to the number 3, the notion “being a human being” has to be mapped to the number 6 [16, ser. VI, vol. 4A, pp. 182, 201-202]. As there is an infinite number of prime numbers, the model is more powerful than it might seem at first glance. In other drafts, Leibniz maps every notion to an ordered pair of a positive and a negative integer [16, ser. VI, vol. 4A, pp. 224-256]. As Leibniz is aware, even with such a characteristica universalis the deduction of an individual statement like “Caesar was murdered on the ides of March” would be impossible, because such a statement involves an infinity of causes and an individual notion like Caesar is composed of an infinity of elements.

The use of numbers for a characteristica universalis even has a metaphysical foundation. Leibniz quotes [16, ser. VI, vol. 4A, p. 263; cf. also ser. I, vol.12, p. 72 and 15, VI, p. 604] the well-known statement [Plato, Philebos 55e; Sapientia Salomonis 11, 21] that God made everything according to measure, number and weight. Admittedly, Leibniz continues, some entities do not have weight, and some entities do not have parts and therefore lack measure. But there is nothing which does not allow for a number. So number is “quasi figura quaedam metaphysica” [16, ser. VI, vol. 4A, p. 264], and arithmetic is therefore a doctrine for the exploration of the powers of things, and thereby the perfection of God’s creation.